Minors and Cofactors of Determinant: Definition, Formula and Solved Examples

Minors and cofactors are one of the most important concepts of the determinants. Minors and Cofactors are important as they help to find out the determinant of large square matrix. The knowledge of Minors and Cofactors helps to determine the adjoint as well the inverse while calculating the determinant of a square matrix. Cofactor Expansion is the term defined for such computation of the determinant.

Read More: Real Numbers Formula

Key Terms: Element of a determinant, square matrix, Minors, Cofactors, Determinants, Cofactor Expansion

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What are Minors?

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Minor is defined as the value calculated from the determinant of a square matrix. It is calculated by crossing the rows and columns corresponding to the given element. Minor of an element such as a_yz of a determinant can be finding out by deleting its yth row and zth column in which the element a_yz lies. Minor of an element a_yz can be represented as M_yz.

Minor of an element of a determinant of the order n(n≥3) is the determinant of the order n-2. 

Example 1.

Find the minor of an element 5 in the determinant:

Solution. Element 5 lies in the 2^nd row and 2^nd column. So, its Minor will be M₂₂. 

M₂₂ =  = 9-21 = -12( obtained by deleting R₂ and C₂ in the ?)

Example 2.

Find the minors of the elements in the determinant:

Solution. Minor of the element M₁₁ = 15 – 12 = 3

Minor of the element M₁₂ = -5 + 8 = 3

Minor of the element M₁₃ = 3 – 6 = -3

Minor of the element M₂₁ = -10 + 12 = 2

Minor of the element M₂₂ = 5 – 8 = -3

Minor of the element M₂₃ = -3 + 4 = 1

Minor of the element M₁₃ = 3 – 6 = -3

Minor of the element M₂₃= -3 + 4 = 1

Minor of the element M₃₃= 3 – 2 = 1

Discover about the Chapter video:

Determinants Detailed Video Explanation:

Read more: Determinant of a Matrix

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What are Cofactors?

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Cofactor is known as the signed minor. Cofactor of an element, a_yz, denoted by A_yz, is defined by A= (-1)^y+zM, where M stand for Minor of a_yz.

Some important rules:

• If the sum of y+z is even, then A_yz = M_yz.
• If the sum of y+z is odd, then A_yz = -M_yz.
• It shows that the difference between the related minors and cofactors are only in the terms of sign.

The value of determinant of order three can be in the terms of minors and cofactors can be defined as:

D= a₁₁M₁₁ - a₁₂M₁₂ + a₁₃M₁₃ or

D= a₁₁C₁₁ - a₁₂C₁₂ + a₁₃ C₁₃

A determinant having order 3 will result in 9 minors and each minor will be a determinant of order 2. Similarly, a determinant of order 4 will have 16 minors, and the determinant will be of order 3.

If cofactor is multiplied to different rows/columns, their sum will be zero.

Example.1: Find minors and cofactors of the determinant:

Solution. Minor of the element ayz is M_yz.

Here, a₁₁= 1. So, minor of M₁₁ = 3

a₁₂ = 2, M₁₂ = 4

 M₂₁ = 2, M₂₂ = 1

Now, cofactor of a_yz is A_yz.

A₁₁=( -1)¹⁺¹ = (-1)²(3)= 3

A₁₂ = (-1)¹⁺² = (-1)³(4) = -4

A₂₁ =( -1)²⁺¹ = (-1)³ (2) = -2

A₂₂ = (-1)²⁺² = (-1)⁴ (1) = 1

Example.2: Find minor and cofactors of the determinant:

Solution. M₁₁ = = -20,  A₁₁ = (-1)¹⁺¹(-20) = -20

M₁₂ = 46 , A₁₂ = -46

M₁₃ = 30 , A₁₃ = 30

M₂₁ = -46 , A₂₁ = 46

M₂₂ = -19 , A₂₂ = -19

M₃₁ = 7 ,  A₃₂ = -7

M₃₃ = -18 ,  A₃₃ = -18

Example.3: Find the cofactor of a11 in the following determinant, D=

Solution. Here a₁₂ = -3, so Minor, M₁₂ = = 42 – 4 = - 38

Cofactor of (-3) will be , A₁₂ = (-1)¹⁺² ( -38) = (-1)(-38) = 38

Also Read:

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Things to Remember

• Minors and cofactors are one of the most important concepts of the determinants.
• Minors and Cofactors are important as they help to find out the determinant of large square matrix.
• The knowledge of Minors and Cofactors helps to determine the adjoint as well the inverse while calculating the determinant of a square matrix.
• Cofactor Expansion is the term defined for such computation of the determinant.
• Cofactor is known as the signed minor.
• Cofactor of an element, a_yz, denoted by A_yz, is defined by A= (-1)^y+zM, where M stand for Minor of a_yz.
• Minor of an element of a determinant of the order n(n≥3) is the determinant of the order n-2. 
• If the sum of y+z is even, then A_yz = M_yz.
• If the sum of y+z is odd, then A_yz = -M_yz.
• It shows that the difference between the related minors and cofactors are only in the terms of sign.

Read more: Symmetric Matrix